Toeplitz Hermitian Positive Definite (THPD) matrices play an important role in signal processing and computer graphics and circular models, related to angular / periodic data, have vide applications in various walks of life. Visualizing a circular model through THPD matrix the required computations on THPD matrices using single bordering and double bordering are discussed. It can be seen that every tridiagonal THPD leads to Cardioid model.
Q-Factor General Quiz-7th April 2024, Quiz Club NITW
On Computations of THPD Matrices
1. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014
73
ON COMPUTATIONS OF THPD MATRICES
Dr. S.V.S. Girija1
, A.V. Dattatreya Rao2
and V.J.Devaraaj3
Associate Professor of Mathematics, Hindu College, Guntur -522002 INDIA
Professor of Statistics, Acharya Nagarjuna University,Guntur-522510 INDIA.
Department of B S & H, V.K.R, V.N.B and A.G.K. College of Engineering, Gudivada,
INDIA.
ABSTRACT
Toeplitz Hermitian Positive Definite (THPD) matrices play an important role in signal processing and
computer graphics and circular models, related to angular / periodic data, have vide applications in
various walks of life. Visualizing a circular model through THPD matrix the required computations on
THPD matrices using single bordering and double bordering are discussed. It can be seen that every
tridiagonal THPD leads to Cardioid model.
1. INTRODUCTION
Every Toeplitz Hermitian Positive Definite matrix (THPD) can be associated to a Circular model
through its characteristic function [Mardia (1972)]. The idea of THPD matrix, a special case of
Toeplitz matrix has a natural extension to infinite case as well. Here, it is attempted to present
certain algorithms for computations on THPD matrices.
Section 2 is devoted to explain the association between Circular models and THPD matrices
which is the motivation for taking up computations on THPD matrices for possible future
applications / extensions. On the lines of Rami Reddy (2005), methods for LU decomposition of
a THPD matrix using single bordering algorithms are discussed and are presented in Sections 3.
2. REPRESENTATION OF A CIRCULAR MODEL THROUGH THPD MATRIX
In continuous case the probability density function (pdf) (
g θ ) of a circular distribution exists
and has the following basic properties
• (
g θ ) θ
∀
≥ ,
0 (2.1)
• 1
)
(
2
0
=
∫ θ
θ
π
d
g (2.2)
• )
2
(
)
( π
θ
θ k
g
g +
= for any integer k (i.e., g is periodic) (2.3)
Rao and Sengupta (2001).
2. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014
74
Distribution function G on the circle is uniquely determined by characteristic function [c.f. p. 80
Mardia (1972)]. The characteristic function and cdf of the resultant wrapped circular model are,
( ) ( ) ( ), 0, 1,
ip
W p E e p p
θ
φ φ
= = = ± 2, 3,...
± ±
and
0
( ) ( ) , [0,2 )
G g d
θ
θ θ θ θ π
′ ′
= ∈
∫
Consider the sequence { }∞
∞
−
p
φ associated with a characteristic function φ of a circular
distribution. It is known that if ∑
∞
−∞
=
p
p
2
φ is convergent then { } 2
l
p ∈
φ and ∑
∞
−∞
=
p
p
2
φ is
convergent iff ∑
∞
=0
2
p
p
φ is convergent. Thus the sequence { }
p
φ of a circular model satisfies (i)
,
1
0 =
φ (ii) p
p φ
φ =
− , ∑
∞
∞
−
∞
<
2
p
φ . Since 0
lim =
p
φ , one can assume that p
p 1 ∀
<
φ .
If p
p
p iβ
α
φ +
= , then the pdf of the corresponding Circular model is given by
( )
+
+
= ∑
∞
=1
sin
cos
2
1
2
1
)
(
p
p
p p
p
g θ
β
θ
α
π
θ (2.4)
Further if )
(θ
g is a function of bounded variation then the Fourier series
∑
∞
−∞
=
−
=
p
ip
p e
g θ
φ
π
θ
2
1
)
( converges [ Jordan’s test ] provided
1. )
(θ
g = { }
)
0
(
)
0
(
2
1
−
+
+ θ
θ g
g or
2. )
(θ
g has only a finite number of maxima and minima and a finite number of
discontinuities in the interval [ )
π
2
,
0 and it can be proved that
∫
=
=
π
θ
θ
θ
φ
2
0
)).
(
(
)
( G
d
e
e
E ip
ip
p
Thus the characteristic function of a Circular distribution G is represented by the sequence { }
p
φ .
Let the matrix )
( j
i
a
A = where 0
,
, ≥
= − j
i
a j
i
j
i φ and i
j
j
i a
a = . Since
p
p φ
φ =
− and all leading principal minors of A are positive, therefore, A is positive definite.
Thus by invoking the Inversion theorem for characteristic functions it follows that for every
Circular model there corresponds an infinite matrix through it’s characteristic function. Further,
3. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014
75
all the leading principal minors are positive, hence the matrix A is THPD matrix. If
0
=
= p
p β
α for n
p ≥ , i.e.
( )
+
+
= ∑
=
n
p
p
p p
p
g
1
sin
cos
2
1
2
1
)
( θ
β
θ
α
π
θ , then the corresponding matrix becomes finite
THPD matrix.
By definition a matrix A is tridiagonal from Girija (2004) if 2
whenever
0 ≥
−
= j
i
a j
i .
When A is a tridiagonal THPD, then as shown below A induces a Cardioid distribution for
appropriate parameters. In view of the involvement of tridiagonal matrices in Circular
distributions particularly in Cardioid distribution we may as well look into the properties
preserved by the tridiagonal matrices in general. We briefly present a few computational methods
connected with tridiagonal matrices.
2.1Cardioid distribution and tridiagonal matrices
The pdf of a Cardioid distribution with parameters ρ
µ and is given by
( ) [ )
π
θ
µ
θ
ρ
π
θ 2
,
0
,
cos(
2
1
2
1
)
( ∈
−
+
=
g ,
2
1
2
1
<
<
− ρ
The trigonometric moments of g are
µ
ρ
α cos
2
1 = , µ
ρ
β sin
2
1 = .
The corresponding characteristic function of the Cardioid distribution is
p
p
W i
p β
α
φ +
=
)
( , 1
,
0 ±
=
p .
Hence, the THPD matrix induced by the characteristic function of a Cardioid distribution is
=
0
1
1
0
φ
φ
φ
φ
A
Clearly for any 2
≥
n , the tridiagonal THPD matrix of order n
=
0
1
0
1
1
0
1
1
0
...
...
...
...
...
...
...
...
0
...
0
0
...
0
...
0
φ
φ
φ
φ
φ
φ
φ
φ
φ
n
A
represents a Cardioid distribution. However the minimum order of such matrix is 2.
4. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014
76
3. LU – DECOMPOSITION OF A TOEPLITZ POSITIVE DEFINITE MATRIX BY
SINGLE BORDERING
Unless otherwise specified, the matrices mentioned here by default are square matrices of order n.
By a LU-decomposition of a matrix A we mean a decomposition of A as
U
L
LU
A and
where
= are lower triangular matrices respectively. This decomposition is
possible [Jain and Chawla (1971)] when the leading submatrices of A are nonsingular.
NOTE :
1. A matrix may not have LU – decomposition as is evident from the following.
Example 1 :
=
3
2
1
3
2
1
0
0
0
3
2
0
u
u
u
l
l
l
2
,
3
,
0 2
1
1
2
1
1 =
=
=
⇒ u
l
u
l
u
l which is impossible.
2. Even if LU decomposition exists, it may not be unique.
Example 2.
=
−
−
1
-
0
0
2
-
1
0
4
-
1
1
-
2
-
6
3
-
0
4
2
-
0
0
1
2
3
3
0
2
2
4
1
1
=
1
0
0
2
1
-
0
4
1
-
1
2
6
-
3
0
4
-
2
0
0
1
-
3. It is known that the LU decomposition is unique when all the entries on the diagonal of L or
U are unity.
Having established the association between a THPD matrices and Circular models, the following
computational aspects on THPD matrix are explored.
LU- decomposition of a THPD matrix by single bordering.
Theorem 3.2
Let A be a THPD matrix of order n and have the form
5. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014
77
=
−
−
−
0
*
1
1
1
a
x
x
A
A
n
n
n
where 1
−
n
A is a nonsingular matrix of order (n-1) and
( )
1
2
1
*
1 ...
... a
a
a
x n
n
n −
−
− = .
Assume that
LU-decomposition of 1
−
n
A exists and is given by 1
1
1 −
−
− = n
n
n U
L
A , then LU
A = where
=
−
−
nn
n
n
l
l
O
L
L
'
1
1
and
= −
−
1
1
1
O
u
U
U n
n
1
1
0
1
1
*
1
1
1
1
1
1 '
,
'
, −
−
−
−
−
−
−
−
−
− −
=
=
= n
n
nn
n
n
n
n
n
n u
l
a
l
U
x
l
x
L
u
Proof: Since 1
−
n
A is THPD matrix 1
1
−
−
n
A exists, hence 1
1, −
− n
n U
L are invertible.
+
=
=
−
−
−
−
−
−
−
−
−
−
−
−
nn
n
n
n
n
n
n
n
n
n
n
nn
n
n
l
u
l
U
l
u
L
U
L
O
u
U
l
l
O
L
LU
1
1
1
1
1
1
1
1
1
1
1
1
'
'
1
'
= A
a
x
x
A
n
n
n
=
−
−
−
0
*
1
1
1
⇔ ,
, 1
1
1
1
1
1 −
−
−
−
−
− =
= n
n
n
n
n
n x
u
L
A
U
L 1
1' −
− n
n U
l =
*
1
−
n
x ,
nn
n
n l
u
l +
−
− 1
1' = 0
a
)
1
(
1
1
1
0
)
1
(
1
1
1
1
1
0 '
' −
−
−
−
−
−
−
−
−
−
−
− −
=
−
=
⇔ n
n
n
n
n
n
n
nn x
A
x
a
x
L
U
x
a
l
( ) 1
*
1
1
1
1
1
*
1 −
−
−
−
−
−
− =
⇔
=
⇔ n
n
n
n
T
n
n l
U
x
U
l
x
( )
( ) 1
*
1
1
1
1
1
*
1
1
1
1
−
−
−
−
−
−
−
−
−
−
=
=
n
n
n
n
n
n
n
n
l
U
L
u
l
U
L
u
For the purpose of computations, the above theorem is arranged in the form of a
Recursive algorithm.
Given A = (ai,j)
Let i
i a
a =
−
6. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014
78
Step 1 :A1 = (a0), L1 = (a0), U1 = (1)
Step 2 :For i = 2 (1) n we have
and ( )
1
2
1
*
1 ...
,
, a
a
a
x i
i
i −
−
− =
Compute
1
1
1
1 , −
−
−
− i
i U
L
Write ,
1
1
*
1
1
−
−
−
− = i
i
T
i U
x
l 1
1
1
1 −
−
−
− = i
i
i x
L
u
Compute 1
1 −
− i
T
i u
l
Write 1
1
0 −
−
−
= i
T
i
ii u
l
a
l
Write ,
0
1
1
=
−
−
ii
T
i
i
i
l
l
L
L
= −
−
1
0
1
1 i
i
i
u
U
U
Write L = Ln, U = Un
Example
We find the LU decomposition for the following THPD matrix A is associated with the
characteristic function of a Wrapped Weibull distribution.
−
−
−
−
+
−
−
+
+
−
+
−
+
+
=
0000
.
1
6347
.
0
5432
.
0
5312
.
0
0276
.
0
2970
.
0
1012
.
0
6347
.
0
5432
.
0
0000
.
1
6347
.
0
5432
.
0
5312
.
0
0276
.
0
5312
.
0
0276
.
0
6347
.
0
5432
.
0
0000
.
1
6347
.
0
5432
.
0
2970
.
0
1012
.
0
5312
.
0
0276
.
0
6347
.
0
5432
.
0
0000
.
1
i
i
i
i
i
i
i
i
i
i
i
i
A
Step1 :
A1 = (1), L1 = (1), U1 = (1)
Step 2:
x1 = [0.5432 - 0.6347i]
1
1
−
L = (1), 1
1
−
U = (1)
l1 = 0.5432 - 0.6347i
u1 = 0.5432 + 0.6347i
l22 = 0.3021
L2 =
0.3021
0.6347i
-
0.5432
0
1.0000
![Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014
73
ON COMPUTATIONS OF THPD MATRICES
Dr. S.V.S. Girija1
, A.V. Dattatreya Rao2
and V.J.Devaraaj3
Associate Professor of Mathematics, Hindu College, Guntur -522002 INDIA
Professor of Statistics, Acharya Nagarjuna University,Guntur-522510 INDIA.
Department of B S & H, V.K.R, V.N.B and A.G.K. College of Engineering, Gudivada,
INDIA.
ABSTRACT
Toeplitz Hermitian Positive Definite (THPD) matrices play an important role in signal processing and
computer graphics and circular models, related to angular / periodic data, have vide applications in
various walks of life. Visualizing a circular model through THPD matrix the required computations on
THPD matrices using single bordering and double bordering are discussed. It can be seen that every
tridiagonal THPD leads to Cardioid model.
1. INTRODUCTION
Every Toeplitz Hermitian Positive Definite matrix (THPD) can be associated to a Circular model
through its characteristic function [Mardia (1972)]. The idea of THPD matrix, a special case of
Toeplitz matrix has a natural extension to infinite case as well. Here, it is attempted to present
certain algorithms for computations on THPD matrices.
Section 2 is devoted to explain the association between Circular models and THPD matrices
which is the motivation for taking up computations on THPD matrices for possible future
applications / extensions. On the lines of Rami Reddy (2005), methods for LU decomposition of
a THPD matrix using single bordering algorithms are discussed and are presented in Sections 3.
2. REPRESENTATION OF A CIRCULAR MODEL THROUGH THPD MATRIX
In continuous case the probability density function (pdf) (
g θ ) of a circular distribution exists
and has the following basic properties
• (
g θ ) θ
∀
≥ ,
0 (2.1)
• 1
)
(
2
0
=
∫ θ
θ
π
d
g (2.2)
• )
2
(
)
( π
θ
θ k
g
g +
= for any integer k (i.e., g is periodic) (2.3)
Rao and Sengupta (2001).](https://image.slidesharecdn.com/1314mathsj06-220127085947/85/on-computations-of-thpd-matrices-1-320.jpg)
![Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014
74
Distribution function G on the circle is uniquely determined by characteristic function [c.f. p. 80
Mardia (1972)]. The characteristic function and cdf of the resultant wrapped circular model are,
( ) ( ) ( ), 0, 1,
ip
W p E e p p
θ
φ φ
= = = ± 2, 3,...
± ±
and
0
( ) ( ) , [0,2 )
G g d
θ
θ θ θ θ π
′ ′
= ∈
∫
Consider the sequence { }∞
∞
−
p
φ associated with a characteristic function φ of a circular
distribution. It is known that if ∑
∞
−∞
=
p
p
2
φ is convergent then { } 2
l
p ∈
φ and ∑
∞
−∞
=
p
p
2
φ is
convergent iff ∑
∞
=0
2
p
p
φ is convergent. Thus the sequence { }
p
φ of a circular model satisfies (i)
,
1
0 =
φ (ii) p
p φ
φ =
− , ∑
∞
∞
−
∞
<
2
p
φ . Since 0
lim =
p
φ , one can assume that p
p 1 ∀
<
φ .
If p
p
p iβ
α
φ +
= , then the pdf of the corresponding Circular model is given by
( )
+
+
= ∑
∞
=1
sin
cos
2
1
2
1
)
(
p
p
p p
p
g θ
β
θ
α
π
θ (2.4)
Further if )
(θ
g is a function of bounded variation then the Fourier series
∑
∞
−∞
=
−
=
p
ip
p e
g θ
φ
π
θ
2
1
)
( converges [ Jordan’s test ] provided
1. )
(θ
g = { }
)
0
(
)
0
(
2
1
−
+
+ θ
θ g
g or
2. )
(θ
g has only a finite number of maxima and minima and a finite number of
discontinuities in the interval [ )
π
2
,
0 and it can be proved that
∫
=
=
π
θ
θ
θ
φ
2
0
)).
(
(
)
( G
d
e
e
E ip
ip
p
Thus the characteristic function of a Circular distribution G is represented by the sequence { }
p
φ .
Let the matrix )
( j
i
a
A = where 0
,
, ≥
= − j
i
a j
i
j
i φ and i
j
j
i a
a = . Since
p
p φ
φ =
− and all leading principal minors of A are positive, therefore, A is positive definite.
Thus by invoking the Inversion theorem for characteristic functions it follows that for every
Circular model there corresponds an infinite matrix through it’s characteristic function. Further,](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)